Sorry, Birdman, but believability and plot hole are two completely different things.
There are hundreds of reasons why the kidnappers might have/want to broadcast live, but it's up to harmonica to write the best one.
If I saw a movie where a bank heister suddenly took off his mask, I'm not thinking "the writer screwed up!"... I'm thinking "what is the writer trying to say about this particular character that he doesn't care if his face is shown?"
And in the case of the wiki definition, illogical means that it doesn't make sense in a literal way i.e. the fact that a particular occurrence negates another occurrence, making it illogical in that sequence of events. Literal, and not subjective. You should have highlighted the section that says "unlikely behaviour or actions of characters" which is subjective, but I wouldn't use a publicly edited website to define a word anyway.
...No, I'm going to have to blatantly disagree with you on this issue. If you have your characters do something that affects the overall plot, and they do so in a way that is "illogical or stupid" ...you've clearly got yourself a "plot hole".
It's a plot hole because you've typed yourself into a corner. You've come up with a plot strategy that is flawed by its own nature and the only way out is to have your characters do something that defies all reason and expectation. You are FORCED to have your characters behave in a stupid way because the overall plot design is flawed.
It has nothing to do with "what the writer is trying to say about a character". The ONLY thing the writer is saying is that he didn't have a good way out of a particular situation, he's stuck in a hole and that he hopes you never discover it.
PLOT HOLES:
Spider-Man 3 (2007): Flint Marko has his whole body turned into sand including his belt buckle and any other metal he happens to be wearing. If all of this turned into sand then why was the locket fine without it turning into sand as well?
The Bourne Identity (2002): In Paris, why does the assassin go to the ridiculous amount of trouble of swinging into the room on a rope with a machine gun when he came in from the lobby (as proved by the dead woman downstairs)? Alternatively, if his intention was always to surprise Bourne by coming in through the window, why venture downstairs at all?
The above two examples are "plot holes". Spider Man was a plot hole based on technical illogic with the inconsistency being in the behavior of the metals. The Bourne Identity's plot hole was based on the illogical actions of the assassin.
A plot hole is anything that has someone questioning the merits of the movie based on certain happenings. If you have a character who develops a complicated bank robbery scheme, plots out every detail, and then removes his mask at a completely stupid time ...the viewer doesn't think, "Gee, I wonder what is going on with this character to make him do something this stupid"? They think to themselves, "Shit, I can't believe I paid $15 for this piece of shit movie".
I'm sorry, ...but that's reality.
-Birdman